The Imaginary Bedrock of Reality
Since its inception in the 1920s, quantum mechanics has been the most successful theory for describing the universe at the scale of atoms and subatomic particles. Its central equations, like the famous Schrödinger equation, have always included a peculiar
element: the imaginary number 'i', the square root of -1. These numbers, called 'complex' when combined with a 'real' number, are used to describe the wave function of a particle, which contains all the information about its possible states. Physicists, including Schrödinger himself, were initially uneasy about a 'fictional' number being so fundamental to describing reality, but they proved so effective and mathematically convenient that they became an accepted, non-negotiable part of the theory.
Challenging a Century of Dogma
The debate over whether complex numbers were just a convenient shortcut or a true feature of nature simmered for decades. In 2021, a landmark study and subsequent experiments seemed to settle the question, concluding that a version of quantum mechanics built only with real numbers would make incorrect predictions about the real world. This appeared to prove that complex numbers were essential. However, that conclusion rested on a key assumption about how to mathematically combine multiple quantum systems. New research, published in mid-2026, challenged that very assumption. A team led by researchers from Heinrich Heine University Düsseldorf and the German Aerospace Center found a way around the problem.
A Different Rule for Reality
The previous work that seemed to lock in complex numbers used a standard mathematical rule known as the tensor product. This rule works perfectly for combining systems in complex-number quantum mechanics. The breakthrough of the new research was realizing this wasn't the only option. The physicists developed a different rule for combining quantum systems, one built on the principle of locality—the intuitive idea that an action in one place shouldn't instantly affect a separate, distant system. By replacing the standard tensor product with this new, physically motivated approach, they successfully constructed a quantum theory using only real numbers.
Identical Predictions, New Perspective
Crucially, this new real-number formulation is not just a mathematical curiosity. It has been shown to make all the same predictions as standard quantum mechanics. Whether describing the spooky action of entanglement or the wave-like behaviour of particles, the real-number theory arrives at the same experimental conclusions as its complex-number counterpart. According to Professor Dagmar Bruß, one of the study's authors, this means the two frameworks are indistinguishable from an experimental standpoint. The use of complex numbers is therefore not fundamental, but a matter of convenience that can be replaced.
What This Means for Physics
This discovery doesn't mean physicists will stop using complex numbers. They remain an incredibly powerful and elegant tool for calculation. However, settling this long-standing foundational question is a major conceptual leap. It clarifies what is truly necessary for a theory of the quantum world versus what is simply useful notation. By showing that the strange, almost magical properties of quantum mechanics can be described without resorting to 'imaginary' numbers, the research provides a new perspective on the mathematical structure of reality. It doesn't change the outcome of quantum experiments, but it profoundly deepens our understanding of the rulebook that governs them.
